SOLVING AN AGGREGATE PRODUCTION PLANNING PROBLEM BY FUZZY BASED GENETIC ALGORITHM (FBGA) APPROACH by IJFLS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

SOLVING AN AGGREGATE PRODUCTION PLANNING PROBLEM BY FUZZY BASED GENETIC ALGORITHM (FBGA) APPROACH Ripon Kumar Chakrabortty1 & Md. A. Akhtar Hasin2 1

Department of Industrial & Production Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh ripon_ipebuet@yahoo.com

Department of Industrial and Production Engineering, Bangladesh University of Science and Technology (BUET), Dhaka-1000, Bangladesh aahasin@ipe.buet.ac.bd

ABSTRACT This paper work demonstrated an interactive Fuzzy Based Genetic Algorithm (FBGA) approach for solving a two products & two periods aggregate production planning (APP) with some vulnerable managerial constraints like imprecise demands, variable manufacturing costs etc. This proposed approach used the strategy of simultaneously minimizing the most possible value, the most pessimistic value & also most optimistic value of the imprecise total costs in lieu of some strong resource constraints. Another important purpose of this study is to derive & observe the variations along with the scope of the imprecise total cost, maximizing the possibility of obtaining lower total costs and also minimizing the risk of obtaining higher total costs. Here the authors employed different unique genetic algorithm parameters scrupulously for solving nondeterministic polynomials problems like APP problems. For reinforcing & accelerating the decision making for the decision maker a case study was considered in a Ready Made garment company in Bangladesh. Consequently, the proposed FBGA approach yields an efficient APP compromise solution and could be efficient for large scale problems.

KEYWORDS Multi-objective optimization, Fuzzy Logic, Genetic Algorithm, Aggregate Production Planning

1. INTRODUCTION Customer demand enters into the production system as units of products. However, production has to be planned as hours of machining and worker hours. In recent era since modernization grasp all kinds of traditional thinking as well as modern manufacturing practices. Obsolescence is the burning issue which takes only a tiny period for any consumer goods. Product life cycles are being shortened with an alarming rate which results unpredicted customer demand. For any supply chain profitability it is mandatory to maintain a balance between supply chain responsiveness & supply chain efficiency. For achieving strategic fit a manufacturing firm bears DOI : 10.5121/ijfls.2013.3101

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

some important roles while acting as a member in any particular multi-echelon supply chain network. The outburst of all these is that the demands are becoming imprecise. So the ultimate goal for any manufacturing firm is to maintain a reach aggregate production planning which encompasses all levels of production costs, inventory costs, sub ordering costs & hiring costs. The primary output of the aggregate planning process is a master schedule, which describes the number of units to be produced during each period and the work force levels required by that period. Aggregate production planning has lured a significant academic researchers & practitioners because of its immense importance. Shorten product life cycle in market & fickle customer perceptions push the researchers to choose this broad area to research (Shi and Haase, 1996). Since the relevant information regarding customer demands are imprecise in nature by using deterministic models bear suboptimal results. By Using fuzzy models such drawback could be terminated (Aliev, 2007). APP problems exaggeratedly imply constraint optimization and since the objective function bears some uncertainty so constrained optimization is must there. Along with fuzzy approach genetic approach also enticed a lot of researchers. Constrained optimizations are often being solved by different direct & indirect approaches. Among indirect approaches genetic approach is mostly lucrative because of its consistent & optimized results (Ioannis, 2009). Genetic algorithm is furnished with different genetic parameters like crossover, mutation, selection functions etc and different researchers used different combinations to solve constrained & unconstrained optimization problems (Bunnag & Sun, 2005). Not only the constrained optimization problem but also Pareto optimizations problems becoming a common field of research (Cai & Wang, 2006). Here the concerned multi-objective problem is also a constrained problem which encompasses all the relevant operating costs, production costs etc. These problems have long attracted the attention of researchers using traditional techniques of optimization and search as well as Genetic algorithms (Schaffer, 1985). Yeh and Chuang (2011) used multiobjective genetic algorithm for partner selection in green supply chain problems. Mixed integer linear programming approach, auxiliary multi-objective problems all utilized substantives results for multi period & multi product aggregate production planning problems (Ramezanian, 2012). Due to NP-hard class of APP, implementation of genetic algorithm and Tabu search are also very common in modern literature for solving these APP problems. Throughout the review it is obvious that there already have been a long evolution phase past for Genetic algorithms as well as fuzzy logic for solving APP problems. Yet the researchers obstinately keep on this & they got newer dimension. Here the authors maintain their optimism while reviewing all the literatures since there have a lot of space for future contributions. Here the authors considered multiple objectives for multi period & multi product APP problem. But the distinction lies in the followed approach. Since the present situation of demand patterns as well as manufacturer’s capacity are changing every day the authors used the imprecise values with a fuzzy triangular membership functions. Again the authors used three distinct scenarios simultaneously with different genetic algorithm options for solving multiple objectives of fuzzy linear programming models including escalating factors. This work develops a novel interactive FBGA approach considering escalating factors as well. The proposed approach used the genetic algorithm options first for solving imprecise multiple objectives & then they are being fuzzified to get optimal solution regarding decision maker’s satisfaction. The rest of this paper is organized as follows: Section 2 describes the problem, details the assumptions, and formulates the problem. Section 2 also focused on the parameters of Auxiliary Multiple Objectives Fuzzy Linear Programming (MOFLP) approach, crisp formulation for solving that model & the ways of 2

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

solving. The following section 3 outlined the FBGA model for the APP case and considered parameters for solving that approach. In section 4 some discussions is placed on a detailed case study which was performed from Comfit Composite Knit Limited (CCKL) for justifying the feasibility of the proposed FBGA approach which is followed by the results & findings in section 5. Conclusions are finally drawn in Section 6.

2. PROBLEM FORMULATION 2.1 Problem description & notation Here in this paper the authors assumed that any particular company manufactures N types of products to satisfy the market demand over a planning horizon T. Generally, the environmental coefficients and related parameters are uncertain in a medium time horizon. Therefore, the forecast demand, related operating costs, and labor and machine capacity are imprecise over the planning horizon. Assigning a set of crisp values for the environmental coefficients and related parameters is inappropriate for dealing with such ambiguous APP decision problems. This APP problem focuses on developing an interactive Fuzzy Based Genetic Algorithm (FBGA) approach to determine the optimum aggregate plan for meeting forecast demand by adjusting regular and overtime production rates, inventory levels, labor levels, subcontracting and backordering rates, and other controllable variables. In addition of that some important assumptions are that the values of all concerned parameters are constant throughout the concerning periods. The forecasted demand over a particular period can be either satisfied or backordered, but the backorder must be fulfilled in the next period. The authors employed the following notations for formulating the APP problem which predominantly akin in different literatures (R. C. Wang & T.F. Liang, 2004; Masud and Hwang, 1980; Wang and Fang, 2001).

Forecasted demand for nth product in period t (units) Regular time production cost per unit for nth product in period t (Tk./unit) Regular time production for nth product in period t (units) Escalating factor for regular time production cost (%) Overtime production cost per unit for nth product in period t (Tk./unit) Overtime production for nth product in period t (units) Escalating factor for overtime production cost (%) Subcontracting cost per unit of nth product in period t (Tk./unit) Subcontracting volume for nth product in period t (units) Escalating factor for subcontract cost (%) Inventory carrying cost per unit of nth product in period t (Tk./unit) Inventory level in period t for nth product (units) Escalating factor for inventory carrying cost (%) Backorder cost per unit of nth product in period t (Tk./unit) Backorder level for nth product in period t (unit) Escalating factor for backorder cost (%) Hiring Cost for one worker in period t (Tk./man-hour) Worker hired in period t (man-hour) Layoff cost for one worker in period t (Tk./man-hour) 3

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

Workers laid off in period t (man-hour) Escalating factor for hire and layoff cost (%) Hours of labor per unit of nth product in period t (man-hour/unit) Hours of machine usage per unit of nth product in period t (machine-hour/unit) Warehouse spaces per unit of nth product in period t (ft2/unit) Maximum labor level available in period t (man-hour) Maximum machine capacity available in period t (machine-hour) Maximum warehouse space available in period t (ft2)

2.2 Fuzzy Based Genetic Algorithm (FBGA) Model 2.2.1 Multi-Objective functions Most practical decisions made to solve APP problems usually consider total costs. Here the authors targeted three objective functions for which was solved by the proposed Fuzzy Based Genetic Algorithm (FBGA). First one selected is total costs as objective function (Masud and Hwang, 1980; Saad, 1982; Wang and Fang, 2001). The total costs are the sum of the production costs and the costs of changes in labor levels over the planning horizon T. Accordingly, the objective function of the proposed model is as follows: .

! # $ % %& '1 ) * ) '1 ) * ) '1 ) * ) '1 ) * ,- ,-

) '1 )

* /

) %' ) * 01 ) 1 ,-

'2*

Here the first five terms are used to calculate production costs. The production costs include five components-regular time production, overtime, and subcontracts, carrying inventory and backordering cost. The later portion specifies the costs of change in labor levels, including the costs of hiring and lay off workers. Escalating factors were also included for each of the cost categories. 2.2.2 Constraints Constraints on carrying inventory:

5 67 89, 8; ' 3-* 4 ' 3-* ) ) ) 4 ) $

'<*

= denotes the imprecise forecast demand of the nth product in period t. In real-world Where, = cannot be obtained precisely in a dynamic APP decision problems, the forecast demand market. The sum of regular and overtime production, inventory levels, and subcontracting and backorder levels essentially should equal the market demand, as in first constraint Equation. Demand over a particular period can be either met or backordered, but a backorder must be fulfilled in the subsequent period. Constraints on Labor levels: 4

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013 .

% ' 3-* 0 ' 3-* ) ' 3-* 1 ) 4 4 % ' ) * $ 0 67 8; .

% ' ) * @

67 8;

'?* 'A*

Here in the fourth constraint equation represents a set of constraints in which the labor levels in period t equal the labor levels in period t-1 plus new hires less layoffs in period t. Actual labor levels cannot exceed the maximum available labor levels in each period, as in fifth equation. Maximum available labor levels are imprecise, owing to uncertain labor market demand and supply. Constraints on Machine capacity & Warehouse space: .

= 67 8; % ̂ ' ) * @

'C*

∑. ,- @ 67 8;

(6)

= denote the imprecise hours of machine usage per unit of the nth product and Here ̂ and the imprecise maximum available machine capacity in period t, respectively. Above two equations represent the limits of actual machine and warehouse capacity in each period. Nonnegativity Constraints on decision variables are: , , , , , , E 0 67 89, 8;

'F*

2.3 Model Development 2.3.1 Model the imprecise data with triangular possibility distribution This study assumes the decision maker (DM) to have already adopted the pattern of triangular possibility distribution for all imprecise coefficients. Fig. 1 presents the triangular possibility H I distribution of imprecise number G $ 0 , , 1. In practice, a DM can construct the triangular possibility distribution of ãnt based on the three prominent data, such as the most H I pessimistic value ( ), the most possible value ( ) and finally the most optimistic value ( ).

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

Fig 1: The triangular possibility distribution of G

Fig 2: The Imprecise data with triangular possibility distributions

2.4 An Auxiliary Multiple Objective Fuzzy Linear Programming (MOFLP) Model 2.4.1 Strategy for solving the imprecise objective function The imprecise objective function of the FBGA model in the previous section has a triangular possibility distribution. Geometrically, this imprecise objective is fully defined by three prominent points (zp, 0) (zm, 1) and (zo, 0). Because of the vertical coordinates of the prominent points being fixed at either 1 or 0, the three horizontal coordinates are the only considerations. Consequently, solving the imprecise objective requires minimizing zp; zm; and zo simultaneously. That is, the proposed approach simultaneously involves minimizing the most possible value of the imprecise total costs, zm, the most pessimistic value, zp & also most optimistic value (zo) of the imprecise total costs. Which indirectly satisfied the objectives of maximizing the possibility of obtaining lower total costs, (zm _ zp), and minimizing the risk of obtaining higher total costs, (zo _ zm). The last two objectives actually are relative measures from zm, the most possible value of the imprecise total costs. Hence necessitates an auxiliary multiple objective fuzzy linear programming model for solving the objective functions first. Fig. 3 depicts the ways minimizing the unpredictable objective functions.

Fig 3: The Strategy to minimize the total costs

As indicated in Fig. 3, possibility distribution J is preferred to possibility distribution KL. This result is presented below for three new crisp objective functions.

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

9 M- $ M

$ % %& '1 ) * ) '1 ) * )

'1 ) * ,- ,-

) '1 ) * ) '1 ) * / +

) %'N ) *01 ) 1 ,-

'O*

9 MP $ M $ % %Q '1 ) * ) '1 ) * ) '1 ) * H

,- ,H

) '1 ) * ) '1 ) * R +

) %0N ) 101 ) 1 ,-

'S*

I I I '1 ) * ) '1 ) * )

'1 ) * 9 MT $ M $ % %& I

,- ,-

I I ) '1 ) * ) '1 ) * / +

) %'N I ) I *01 ) 1 ,-

'2U*

2.4.2 Crisp formulation Recalling Eq. (2) of the FBGA model the author considered the situation in which available resource (the right hand side), Dnt, is imprecise and has triangular possibility distributions with the most possible and least possible values. In real-world APP decision problems, the decision maker can estimate a possible interval for imprecise demand based on experience and knowledge. The main problem is obtaining a crisp representative number for the imprecise demand. This work adopts the weighted average method proposed by Lai and Hwang (1992) to convert Dnt into a crisp number. If the minimum acceptable possibility, β, is given, then the auxiliary crisp equality constraints can be presented as follows: I ' 3-* 4 ' 3-* ) ) ) 4 ) $ V- ,W ) VP ,W ) VT ,W 89, 8; '22* H

Where w1+w2+w3 = 1, w1, w2, w3 denote the weights of the most pessimistic, most possible and most optimistic value of the imprecise demand, respectively. The weights w1, w2 and w3 can be determined subjectively by the experience and knowledge of the DM. Similarly, if the minimum acceptable possibility, β, is given, then the auxiliary crisp inequality constraints on Eq. (4) can be presented as follows: .

I % ' ) * @ V- ,W ) VP ,W ) VT ,W

'2<*

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

To solve the imprecise right-hand sides in Eq. (11) and (12), this work applies the concept of the most likely values proposed by Lai and Hwang’s (1992) approach, assuming w2 = 4/6 and w1 = w3= 1/6. The reason for using the most likely values here is that the most possible values usually are the most important ones, and thus should be assigned more weights. Moreover, to solve Eq. (5) with imprecise technological coefficient (r#nt) and available resources M#tmax, the approach proposed here converted these imprecise inequality constraints into a crisp one using the fuzzy ranking concept (Tanaka et al., 1984; Lai and Hwang, 1992). Consequently, the auxiliary inequality constraints on Eq. (5) can be presented as follows: .

% ,W ' ) * @ ,W 8;

'2?*

' ) * @ ,W 8; % ,W

'2A*

X X ' ) * @ ,W % ,W 8;

'2C*

2.5 Solving the auxiliary MOFLP problem Furthermore, the auxiliary MOLP problem can be converted into an equivalent single-goal LP problem using the fuzzy decision-making of Bellman and Zadeh (1970) and Zimmermann’s fuzzy programming method (1978). The positive ideal solutions (PIS) and negative ideal solutions (NIS) of the three objective functions can be specified as follows, respectively (Hwang and Yoon, 1981; Lai and Hwang, 1992). z1PIS = Min Zm,

z1NIS = Max Zm

(16a)

z2PIS = Max (Zm-Zo),

z2NIS = Min (Zm-Zo)

(16b)

z3PIS = Min (Zo-Zm),

z3NIS = Max (Zo – Zm)

(16c)

For each objective function, the corresponding linear membership function is defined by 1, Y - ^ Y-_à ] [ Y-.à 4 Y _à 6- 0Y - 1 $ .à @ Y - @ Y-.à c _à , YY 4 Y \ [ .à 7, Y Z - b Y-

_à ]1, Y P ^ YP [ Y P3 dfgh 6P 0Y P 1 $ _à e .à , YP.à @ Y P @ YP_à c \YP 4 YP [ 7, Y b Y .à P P Z

'2F*

'2O*

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

1, Y T ^ YT_à ] [ YT.à 4 Y T 6T 0Y T 1 $ .à , YT_à @ Y T @ YT.à c \YT 4 YT_à [ .à Z 7, Y T b YT

'2S*

Here Zf1, Zf2 and Zf3 are the fuzzified objective function values for determining the corresponding linear membership function. Hence to mention that the total cost is imprecise & has a triangular possibility distribution. So, Z¯ = (zf1 – zf2, zf1, zf1 + zf3) ≡ (Z1, Z2, Z3). Using the fuzzy decisionmaking of Bellman and Zadeh (1970) and Zimmermann’s (1978) fuzzy programming method, the complete equivalent single-goal LP model for solving the APP decision problem can be formulated as follows: Max L Subject to,

L @ fk 'zmk *,

i $ 1,2,3

} Ir's3-* 4 Br's3-* ) Q rs ) Ors ) Srs 4 Irs ) Brs $ w- Drs,z ) wP D| rs,z ) wT Drs,z 8n, 8t {

r,-

% ir's3-* 0Q r's3-* ) Or's3-* 1 ) Hs 4 Fs 4 % irs 'Q rs ) Ors * $ 0

| } % irs 'Q rs ) Ors * @ w- Ws| ,z ) wP Ws| ,z ) wT Ws| ,z {

8t,

% rrs,z 'Q rs ) Ors * @ Ms| ,z 8t,

r,-

{

| | 'Q rs ) Ors * @ Ms| ,z % rrs,z 8t,

X X 'Q rs ) Ors * @ Ms| ,z % rrs,z 8t,

% Vrs Irs @ Vs|

r,-

0@L@1

8t,

Q rs , Ors , Srs , Irs , Brs , Hs , Fs E 0

8n, 8t 9

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

3. OUTLINE OF THE BASIC FBGA MODEL The algorithm of the proposed FBGA approach for solving the APP decision problem is as follows. Step 1: Formulate the FBGA model for the multiproduct APP decision problem considering escalating factors on all imprecise parameters. Step 2: Model the imprecise coefficients (ãnt, b#nt, c#nt, d#nt, ẽnt, k#t, m#t, r#nt) and right-hand sides (D#nt, W#t, M#t) using triangular possibility distributions. Step 3: Develop three new crisp objective functions of the auxiliary MOLP problem. Step 4: Given the minimum acceptable possibility, β, convert the imprecise constraints into crisp ones using the weighted average method or the fuzzy ranking concept. Step 5(i): For solving the imprecise multiple objective functions using Genetic Algorithm first of all generate random population of n chromosomes Step 5(ii): Evaluate simultaneously the Multiple fitness Zi of each chromosome x in the population Step 5(iii): Create a new population by repeating four steps (Selection, Crossover, Mutation and Acceptation) until the new population are complete. Step 5(iv): Use new generated population for a further run Step 5(v): If the stopping condition is satisfied, stop, and return the best solution in current population Step 5(vi): If the stopping condition is not satisfied then go to step 2 & follow loop. Step 6: After getting imprecise multiple objective functions, specify the linear membership functions for them, and then convert the auxiliary MOLP problem into an equivalent FLP model using the fuzzy decisions of Bellman and Zadeh (1970) and Zimmermann’s (1978) fuzzy programming method. Step 6: Solve and modify the model interactively. If the DM is not satisfied with the initial solution, then the model must be modified until a satisfactory solution is found.

3.1 Fuzzy Based Genetic Algorithm (FBGA) Model Parameters The mechanics of a simple genetic algorithm are surprisingly simple, involving nothing more complex than copying strings and swapping partial strings. A simple genetic algorithm that yields good results in many practical problems is composed of three operators. Those are Reproduction, Crossover & Mutation. Crossover options specify how the genetic algorithm combines two individuals, or parents, to form a crossover child for the next generation. Here the authors choose five different crossover options for five scenarios. Here the authors employed three kinds og crossover and those are scattered crossover in which if p1 and p2 are the parents such as p1 = [a b c d e f g h] and p2 = [1 2 3 4 5 6 7 8] and the binary vector is [1 1 0 0 1 0 0 0], then the function returns the following child1 = [a b 3 4 e 6 7 8]. Second crossover was the intermediate crossover function and the final one was arithmetic crossover where random weighting factors were considered. Then another genetic algorithm parameter is the mutations which specify how the genetic algorithm makes small random changes in the individuals in the population to create 10

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

mutation children. Here the authors use Constraint dependent mutation & Adapt feasible mutation options. Another important term is creation function. Creation function creates the initial population for genetic algorithm. Here the authors choose feasible population & constraint dependent creation functions. Next parameter is selection function where the authors used only tournament selection option for tournament size 2 & 4.

4. MODEL IMPLEMENTATION 4.1 Case description The Comfit Composite Knit Limited is the sister concern of Youth Group, which is one of the pioneer company of Ready Made Garments (RMG) sector in Bangladesh. This company readily produced knit ware items among them some are fancy & some are expensive. The jacket items as well as cardigan items are most expensive and most time & cost incurring manufacturing items. So it needs a lot of precise observations & perfect manufacturing practices to catch up the market & satisfy the buyers within specified lead time. Since they are the most expensive items, major concentration was on one particular style of hooded jacket (Product 1) & another special type of ladies cardigan (Product 2). The APP decision problem for CCKL’s Knit garments manufacturing plant presented here focuses on developing an interactive Genetic Algorithm approach for minimizing total costs. The planning horizon is 2 months long, including May and June. The model includes two types of knit ware items, namely the hooded jacket (Product 1) and special type of ladies cardigan (Product 2). According to the preliminary environmental information, Tables 1 & 2 summarizes the forecast demand, related operating cost, and capacity data used in the CCKL case. Other relevant data are as follows. I. II. III.

IV.

Initial inventory in period 1 is 500 units of product 1 and 200 units of product 2. End inventory in period 2 is 400 units of product 1and 300 units of product 2. Initial labor level is 225 man-hours. The costs associated with hiring and layoffs are Tk. (20, 22, 25) and Tk. (6, 8, 10) per worker per hour, respectively. Hours of labor per unit for any periods are fixed to 0.033 man-hours for product 1 and 0.05 man hours for product 2. Hours of machine usage per unit for each of the two planning periods are (0.09, 0.1, 0.11) machine-hours for product 1 and (0.07, 0.08, 0.09) machine-hours for product 2. Warehouse spaces required per unit are 1 square feet for product 1 and 1.5 square feet for product 2. The expected escalating factor in each of the costs categories are 1%. Table 1 Forecasted demand, maximum labor, machine and warehouse capacity data Item (Units)

Period 1 2 D#1t (pieces) (1200, 1400, 1500) (2750,3000, 3200) D#2t (pieces) (1400, 1600, 1800) (700, 800, 1000) W#tmax (man-hours) (180, 225, 250) (180, 225, 250) M#tmax (machine(360, 400, 430) (450, 500, 540) Vtmax (ft2) 1000 1000 Table 2 Related Operating cost data for the CCKL case

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

Product

ant (tk./unit)

bnt (tk./unit)

cnt (tk./unit)

dnt (tk./unit)

ent (tk./unit)

(20, 22, 25)

(38, 40, 44)

(25, 27, 30)

(3, 3.5, 4)

(40, 42, 44)

(18, 20, 22)

(36, 40, 42)

(28, 30, 32)

(3.5, 4, 4.5)

(45, 47, 50)

The PIS of the three new objective functions is (Tk.130, 000, Tk.20, 000, Tk.10, 000) and the NIS is (Tk.240, 000, Tk.10, 000, Tk.50, 000). The corresponding linear membership function of the three new objective functions can be defined according to Equations. (17) to (19). The authors used MATLAB computer software to solve the proposed Fuzzy Based Genetic Algorithm (FBGA) approach for the CCKL case. Total Three runs were implemented considering three scenarios with different FBGA parameters shown in Table 3. Table 4 lists the multiple objective values for three FBGA runs through MATLAB. Finally Since from Table 4 it is clear that the least cost is achieved in the third scenario so in Table 6 lists the entire initial APP plan for the CCKL case based on the present information for that third scenario. Here the initial total cost is imprecise and has a triangular possibility distribution of (Tk.2,17,238, Tk.2,35,087, Tk.2,56,457) for the third FBGA run, and overall degree of Decision Maker’s satisfaction with multiple goal values is 0.79. Moreover, the DM may try to modify the results interactively by adjusting the linear membership functions and related model parameters until a satisfactory solution is obtained. Table 3 Different Genetic Algorithm options used for three scenarios FBGA Parameters/ Options Population Type Population Size Creation Function Mutation Crossover Migration (Fraction) Reproduction (Fraction) Selection (Size) Distance Measure Function Pareto Front Pop. Fraction Iteration needed to complete

Scenario 1

Scenario 2

Scenario 3

Double 360 Constraint dependent Constraint dependent Intermediate Forward (0.2) Crossover (0.8)

Double 360 Feasible Population Constraint dependent Scattered Forward (0.2) Crossover (0.8)

Double 360 Constraint dependent

Arithmetic Both (0.5) Crossover (0.5)

Tournament (2)

Tournament (4)

Crowding

0.35

107 Generations

116 Generations

136 Generations

Adapt Feasible

Table 4 Imprecise Objective Function values for different scenarios 12

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

Imprecise Objectives

Scenario 1

Scenario 2

Scenario 3

PIS

NIS

2,17,272.30

2,17,273

2,17,238.30

1,30,000

2,40,000

2,35,121.88

2,35,123.37

2,35,086.98

20,000

10,000

2,56,493

2,56,497.16

2,56,457

10,000

50,000

Total Cost(Tk.)

7,08,886.20

7,08,893.55

7,08,782.32

Table 5 Fuzzified objective values & other values for the APP plan of CCKL case Fuzzifie d Objecti ves Scenario 1 Scenario 2 Scenario 3

zf1

zf2

zf3

f1(zf1)

f2zf2)

f3(zf3)

2,35,120.9

17,848.6

21,372.2

0.044

0.78

0.72

2,35,123.6

17,850.3

21,373.8

0.044

0.79

0.71

2,35,087

17,848.6

21,370

0.044

0.79

0.72

0.79

Minimum Total Cost (Z ) Tk. (217238, 235086,25 6457)

Table 6 Initial multi-product & multi-period APP plan for the CCKL case (Third Scenario) Items (Product 1)

Period

Q1t (Units)

1 544

2 1026

O1t (Units)

544

S1t (Units)

Items (Product 2)

Period

Q2t (Units)

1 664

2 454

1010

O2t (Units)

670

454

186

333

S2t (Units)

I1t (Units)

446

524

I2t (Units)

160

212

B1t (Units)

582

B2t (Units)

140

Ht (man-hours)

102.65

173.40

Ft (man-hours)

163.49

5. RESULTS & FINDINGS The APP decision problem presented in the CCKL case was solved using the crisp ordinary Fuzzy Linear Programming (FLP) model, as summarized in Table 7. Consequently, the optimal value when applying FLP to minimize the total costs was Tk. 2, 35,087. In contrast with the proposed FBGA approach, the improved results were Tk. (217238, 235087, and 256457) for the third scenario. These figures indicate that the FBGA solutions are an efficient compromise solution, compared to the optimal goal value obtained by the FLP model. Thereafter, an improved APP plan is obtained by the proposed FBGA approach under an acceptable degree of DM 13

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

satisfaction in a fuzzy environment. From the initial results and linear membership function values, it is revealed that the changes in the PIS and NIS of the objective functions, influences the objective and L values. These findings imply that the DM must specify an appropriate set of PIS and NIS of the objective functions for making APP decisions, to effectively seek the right linear membership function for each objective function. Finally, the proposed FBGA approach is the most practical for solving APP decision problems and can generate better decisions than other models. The proposed FBGA approach outputs more wide-ranging decision information than other models. The proposed FBGA approach focuses on the multi-periods and multi-products (product family) problems in an APP decision making process. Table 7 Comparison of solutions between FLP & Proposed Approach Item

FLP model

The proposed FBGA approach

Objective function

Min Z

Max L

L (DM’s overall degree of 100% satisfaction)

79% (can be more after changing membership functions)

Z (Total costs)

(Tk. 2,17,238, Tk.2,35,087, Tk.2,56,457)

Tk.2, 35,087

The proposed approach also provides information on alternative strategies for overtime, subcontracting, inventory, backorders, and hiring and layoffs workers, in response to variations in forecast demand. Additionally, the proposed model considers the actual limitations in labor, machine, and warehouse capacity.

6. CONCLUSIONS This work presents a novel interactive FBGA approach for solving multi product and multi period APP decision problems with the forecasted demands, related operating costs, and capacity. The proposed FBGA approach not only provides more computational efficiency and more flexible doctrines, but also supports possibilistic decision-making in an uncertain environment. This proposed FBGA approach also can helps to determine optimum solution even it is NP (nondeterministic polynomial) hard problems. Different Genetic Algorithm options have been considered in this APP problem which could make an impression for the future researchers to choose the suitable combination for solving multiple objective problems. Consequently, the proposed approach is expected to be suitable for making real world APP decisions.

7. REFERENCES [1]

[2] [3]

R.A. Aliev, B. Fazlollahi, B.G. Guirimov & R.R. Aliev, (2007) “Fuzzy-genetic approach to aggregate production–distribution planning in supply chain management”, Information Sciences, Vol. 177, pp. 4241–4255. A. Baykasoglu, (2001) “MOAPPS 1.0: Aggregate production planning using the multiple- objective Tabu search”, International Journal of Production Research, Vol. 39, pp. 3685–3702. R.E. Bellman & L.A. Zadeh, (1970) “Decision-making in a fuzzy environment”, Management Science, Vol.17, pp. 141–164. 14

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013 [4] [5] [6] [7]

[8]

[9] [10] [11] [12]

[13]

[14] [15]

[16]

[17] [18]

[19] [20] [21] [22] [23] [24] [25]

J.J. Buckley, (1988) “Possibilistic linear programming with triangular fuzzy numbers”, Fuzzy Sets and Systems, Vol. 26, pp. 135–138. J.J. Buckley, (1989) “Solving possibilistic linear programming problems”, Fuzzy Sets and Systems, Vol. 31, pp. 329–341. D. Bunnag & M. Sun, (2005) “Genetic algorithm for constrained global optimization in continuous variables”, Applied Mathematics and Computation, Vol. 171, pp. 604–636. Z. Cai & Y. Wang, (2006) “A Multi-objective Optimization-Based Evolutionary Algorithm for Constrained Optimization”, IEEE Transactions on Evolutionary Computation, Vol. 10, No. 6, pp. 658-675. R.Y.K. Fung, J. Tang & D. Wang, (2003) “Multiproduct aggregate production planning with fuzzy demands and fuzzy capacities”, IEEE Transactions on Systems, Man, and Cybernetics—Part A: Systems and Human, Vol. 33, No. 3, pp. 302–313. D.E. Goldberg, (1989) Genetic Algorithms in Search, Optimization & Machine Learning, Pearson Education Pvt. Ltd., Singapore. C.C. Holt, F. Modigliani, H.A. Simon, (1955) “Linear decision rule for production and employment scheduling”, Management Science, Vol. 2, pp. 1–30. G.T. Ioannis, (2009) “Solving constrained optimization problems using a novel genetic algorithm”, Applied Mathematics and Computation, Vol. 208, pp. 273–283. M. Inuiguchi & M. Sakawa, (1996) “Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test”, Fuzzy Sets and Systems, Vol. 78, pp. 231– 241. H.A. Jensen & S. Maturana, (2002) “A possibilistics decision support system for imprecise mathematical programming problems”, International Journal of Production Economics, Vol. 77 (2), pp. 145–158. Y.J. Lai & C.L. Hwang, (1992) “A new approach to some possibilistic linear programming problems”, Fuzzy Sets and Systems. Vol. 49, pp. 121–133. S.M. Masud & C.L. Hwang, (1980) “An aggregate production planning model and application of three multiple objective decision methods”, International Journal of Production Research, Vol. 18, pp. 741–752. R. Ramezanian, D. Rahmani & F. Barzinpour, (2012) “An aggregate production planning model for two phase production systems: Solving with genetic algorithm and Tabu search”, Expert Systems with Applications, Vol. 39, pp. 1256-1263. D.K. Sharma & R.K. Jana, (2009) “Fuzzy goal programming based genetic algorithm approach to nutrient management for rice crop planning”, Int. J. Production Economics, Vol.121, pp. 224–232. V.S. Summanwar, V.K. Jayaraman, B.D. Kulkarni, H.S. Kusumakar, K. Gupta & J. Rajesh, (2002) “Solution of constrained optimization problems by multi-objective genetic algorithm”, Computers and Chemical Engineering, Vol. 26, pp.1481–1492. R.C. Wang, R.C & H.H. Fang, (2001) “Aggregate production planning with multiple objectives in a fuzzy environment”, European Journal of Operational Research, Vol. 133, pp. 521–536. R.C. Wang & T.F. Liang, (2004) “Application of fuzzy multi-objective linear programming to aggregate production planning”, Computers and Industrial Engineering, Vol. 46, No. 1, pp. 17–41. R.C. Wang & T.F. Liang, (2005) “Applying possibilistic linear programming to aggregate production planning”, International Journal of Production Economics, Vol. 98, pp. 328-341. L.A. Zadeh, (1978) “Fuzzy sets as a basis for a theory of possibility”, Fuzzy Sets and Systems, Vol. 1, pp. 3–28. H.J. Zimmermann, (1976) “Description and optimization of fuzzy systems”, International Journal of General Systems, Vol. 2, pp. 209–215. H.J. Zimmermann, (1978) “Fuzzy programming and linear programming with several objective functions”, Fuzzy Sets and Systems, Vol. 1, pp. 45–56. G. Kabir & M.A.A. Hasin, (2011) “Comparative analysis of AHP and Fuzzy AHP models for multicriteria Inventory classification”, International Journal of Fuzzy Logic Systems, Vol. 1, No. 1, pp. 116. 15

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013 Authors Ripon Kumar Chakrabortty is a Lecturer in the Department of Industrial and Production Engineering at Rajshahi University of Engineering and Technology (RUET), Dhaka, Bangladesh. He obtained his B.Sc. Engineering in Industrial and Production Engineering from Bangladesh University of Engineering and Technology (BUET), Dhaka, Bangladesh, and currently he is continuing his M. Sc. Engineering there. His main scientific interests concentrate on linear & nonlinear optimization, mathematical modeling for PSO, GA & Fuzzy. He has approximately 2 years and 7 months of teaching and research experience in Bangladesh. He has in his credit approximately 23 published & accepted international journal and conference papers. Dr. M. Ahsan Akhtar Hasin is a Professor in the Department of Industrial and Production Engineering at Bangladesh University of Engineering and Technology (BUET), Dhaka, Bangladesh. He obtained his B.Sc. Engineering in Electrical and Electronic Engineering from BUET, Dhaka, Bangladesh, M. Sc. Engineering and Ph.D. in Industrial Systems Engineering from the Asian Institute of Technology (AIT), Bangkok, Thailand. He has 24 years of teaching and research experience in Bangladesh, Thailand and Vietnam. He has authored several textbooks and chapters in books, published from the USA, UK and Bangladesh. He has in his credit a large number of international journal publications too. His one of the research publications was awarded the “best research publication” in the world in year 2001.